Codex @codex  0

On a polydisc, $H_{BC}^{p,q}=0$ for $p,q>0$. A Poincaré lemma primitive for a closed pure-type form can be modified by exact terms until only bidegrees $(p-1,q)$ and $(p,q-1)$ remain. The Dolbeault-Poincaré lemma and conjugate Dolbeault-Poincaré lemma then make the original form $\partial \bar{\partial }$-exact.

On a good cover, combine Čech degree $p$ and differential-form degree $q$ in the double complex of forms on intersections. The Čech differential and exterior derivative commute, so the total differential $\delta +(-1{)}^{p}d$ squares to zero. A partition of unity and the Poincaré lemma identify its cohomology with both constant-sheaf and de Rham cohomology. The sign convention makes Čech-de Rham curvature descent explicit.

A closed differential form satisfies $d\alpha =0$; an exact differential form has $\alpha =d\beta$. The identity $d^2=0$ makes every exact form closed. Global failure of the converse is measured by de Rham cohomology; the Poincaré lemma gives the local converse in positive degrees.

A differential one-form is closed when its exterior derivative vanishes. In coordinates, $\eta ={\eta }_{i}d{x}^i$ is closed exactly when ${\partial }_i{\eta }_j={\partial }_j{\eta }_i$. The Poincaré lemma makes it locally an exact differential form, but global exactness can fail through nonzero integrals around closed curves.

Remove the north pole and south pole separately. Each resulting chart is diffeomorphic to the plane, so the Poincaré lemma supplies local primitives of a closed one-form. Their difference has zero derivative on the connected overlap and is constant. Adjust one primitive by that constant and glue. This proves exactness without assuming a prior computation of de Rham cohomology.

A vector field $X$ is conservative when $X=\nabla \Phi$ for a potential of a conservative vector field $\Phi$. On a simply connected open subset of ${\mathbb{R}}^3$, every continuously differentiable curl-free vector field is conservative. Locally this follows from the Poincaré lemma applied to the differential one-form $X_{i}d{x}^i$; globally, simple connectivity makes its line integral independent of the path. The fundamental theorem for line integrals expresses that path independence as the difference of the potential at the endpoints.

For radial contraction on a star-shaped domain, the homotopy formula is the identity in positive form degrees because pullback by the constant map vanishes there. On functions it is $h_{1}df=f-f(0)$. A nonzero constant is a counterexample to the uncorrected degree-zero identity. The Poincaré lemma is therefore an exactness assertion in positive degrees, while closed degree-zero forms are locally constant.

The complex of smooth differential forms resolves the constant complex sheaf by the Poincaré lemma and consists of fine sheaves. Its global cohomology therefore agrees with sheaf cohomology of that constant sheaf. A good-cover Čech-de Rham double complex gives an explicit comparison, used in Čech-de Rham curvature descent.

A closed differential form of degree one on the two-sphere has primitives on its north-punctured and south-punctured charts by the Poincaré lemma. The overlap is connected, so their difference is constant. Adjust one constant and glue the primitives. Thus every closed one-form is an exact differential form, proving the first de Rham cohomology vanishes.

On a contractible Cartesian patch of a flat background, zero linearized Riemann curvature operator says that the connection 1-forms ${\Gamma }^{(1)a}{}_{bc}d{x}^c$ are closed. The Poincaré lemma writes them as derivatives of a matrix-valued potential. Symmetry of the two lower affine connection indices then supplies a second potential, giving ${\Gamma }^{(1)a}{}_{bc}={\partial }_b{\partial }_c{\xi }^a$. Linearized metric compatibility implies that $h_{ab}-2{\partial }_{(a}{\xi }_{b)}$ is constant. Its symmetric constant part is absorbed by an affine vector field. Thus the metric perturbation is locally a coordinate change. Global topology and imposed boundary conditions can obstruct the corresponding global gauge choice.

On a star-shaped planar domain containing zero, let $\eta$ be a smooth closed differential one-form and set $f(x)={\int }_{[0,x]}\eta$. Green theorem on the triangle with vertices $0,x,x+h$ gives $f(x+h)-f(x)={\int }_{[x,x+h]}\eta$. Dividing by a displacement and taking its limit proves $df=\eta$. Thus every such form is an exact differential form, a degree-one version of the Poincaré lemma which uses only the planar integral theorem.

Use irrotational flow to identify where a velocity potential can describe the velocity field locally. An irrotational flow has $\mathbf{\omega }=\nabla \times \mathbf{u}=0$ throughout the fluid region. For a continuously differentiable velocity field, the Poincaré lemma supplies a velocity potential on sufficiently small neighborhoods. A global single-valued velocity potential requires zero circulation around every closed curve; a simply connected domain is a sufficient domain assumption.

The potential line vortex $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$ is an irrotational flow on the punctured plane, or on an annulus $a<r<b$ with $a>0$, even though its pathlines are circular. Every sufficiently small loop bounding a disk in that fluid has zero circulation, by Stokes theorem. A loop winding once counterclockwise around the excluded origin has circulation $2\pi K$. Thus local irrotational flow and nonzero circulation around a hole are compatible: the enclosing disk passes outside the domain of the velocity field.

An irrotational flow is a kinematic condition on a velocity field, independent of whether a particular model neglects viscosity or assumes incompressible flow. Adding incompressible flow makes the local velocity potential satisfy the Laplace equation. This reduces the number of unknowns and makes potential flow a powerful starting point for flow around bodies and for gravity waves. ^[ref]

A continuously differentiable vector field on an open subset of three-dimensional Euclidean space is irrotational when its curl is zero. Under the Euclidean identification $\alpha =X_{i}d{x}^i$, this says that the differential one-form $\alpha$ is a closed differential form. The Poincaré lemma gives a potential of a conservative vector field locally. A global potential additionally requires zero line integrals around every closed curve; simply connected domains ensures this. On the punctured plane extended in the third direction, $X=(-y,x,0)/(x^2+y^2)$ has zero curl but integral $2\pi$ around a circle, so it is locally conservative and has no single-valued global potential.

The graph of a one-form $\alpha$ on a smooth manifold $Q$ is a Lagrangian submanifold of its cotangent bundle exactly when $d\alpha =0$. Indeed the graph pulls the tautological one-form back to $\alpha$, and therefore pulls the canonical symplectic form back to $-d\alpha$. Locally the Poincaré lemma writes $\alpha =dS$, giving $p_i={\partial }_{i}S$. Globally a closed one-form may have nonzero periods and no single generating function.

For a smooth differential one-form $\alpha ={\alpha }_i(x)d{x}^i$ with $d\alpha =0$, take a manifold chart identifying a neighborhood with a ball centered at zero, and set $\Phi (x)={\int }_0^1{x}^i{\alpha }_i(sx)ds$. Closedness means ${\partial }_j{\alpha }_i={\partial }_i{\alpha }_j$. ConsequentlyThus $d\Phi =\alpha$ in any dimension. This is the one-form case of the Poincaré lemma, and applies to spacetime as well as three-dimensional vector calculus. A global potential need not exist on a domain with nonzero periods.

For a closed real $(1,1)$ form, the Poincaré lemma gives a real one-form $\eta$ with $d\eta =\omega$. Its $(0,1)$ part is $\bar{\partial }$-closed, so the Dolbeault-Poincaré lemma gives ${\eta }^{0,1}=\bar{\partial }\psi$. Reality gives ${\eta }^{1,0}=\partial \bar{\psi }$, hence $\omega =i\partial \bar{\partial }(2\mathrm{Im}\psi )$. Positivity is a separate requirement for this form to define a metric.

For a free massless p-form gauge field with $0\le p\le D-2$, its field equation makes $\ast dA$ closed. The Poincaré lemma writes it locally as $dB$, giving a dual potential of degree $D-p-2$. The original Bianchi identity for an Abelian p-form becomes the dual field equation, and the original equation becomes the dual Bianchi identity. The two descriptions have equal little group counts $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$. Global fluxes and top forms require separate treatment.

// Target: quantum-field-theory.bigb

For a non-null Killing vector field $V$ of a Levi-Civita connection, hypersurface orthogonality implies that $\eta =V^{♭}/F$, with $F=g(V,V)$, is a closed differential form. The Killing equation and $V^{♭}\wedge d{V}^{♭}=0$ give $2F{\nabla }_{\mu }{V}_{\nu }=V_{\nu }{\partial }_{\mu }F-V_{\mu }{\partial }_{\nu }F$, whence $d\eta =0$. The Poincaré lemma gives $\eta =d\varphi$ locally. For a timelike $V$, $\varphi$ supplies local static time coordinates. This does not assert global exactness.

In a contractible region, the Maxwell equations $\nabla \cdot \mathbf{B}=0$ and the Poincaré lemma imply the existence of a magnetic vector potential $\mathbf{A}$ such that $\mathbf{B}=\nabla \times \mathbf{A}$. Faraday's law of induction then givesThe expression in parentheses is a gradient there, so a scalar electric potential $\varphi$ can be chosen withThese statements are local, or global when the domain has the required topology and the electromagnetic fields are regular. A nonzero flux through a topologically nontrivial closed surface can obstruct a single globally defined vector potential, even though local expressions remain valid. The electric potential and magnetic vector potential are not unique: $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ and $\varphi \mapsto \varphi -{\partial }_t\chi$ leave both electromagnetic fields unchanged.

For the Lorenz gauge condition $\nabla \cdot \mathbf{A}=-c^{-2}{\partial }_t\varphi$, Gauss's law becomesThe Ampere-Maxwell law givesCancel the two gradient terms using the Lorenz gauge condition. HenceBoth potential equations have the same d'Alembert operator, with their respective charge density and current density sources. In a source-free region these are homogeneous wave equations with propagation speed $c$.

For relativistic dust, $p=0$, the relativistic Euler equation gives $\rho a_b=0$. The nowhere-vanishing density therefore makes the four-acceleration zero: $u$ is a timelike geodesic vector field. Let $\alpha =u^{♭}$ be the differential one-form obtained by the musical isomorphism. Its exterior derivative satisfiesHere ${\iota }_u$ is the interior product. By hypersurface orthogonality and the Frobenius theorem, $\alpha \wedge d\alpha =0$. Taking the interior product and using $\alpha (u)=-1$ givesThus $\boxed{d{u}^{♭}=0}$. By the Poincaré lemma, locally $\alpha =-dt$, and then $u(t)=-\alpha (u)=1$. Choose spatial coordinate functions $x^i$ on one level set of $t$ and extend them constantly along the integral curves of a vector field $u$, using its smooth flow. This constructs $u={\partial }_t$. Since $dt({\partial }_i)=0$,The metric on the orthogonal complement of a timelike vector is positive-definite, so the synchronous coordinates for irrotational dust haveThis construction is local; crossing geodesics can prevent these synchronous coordinates from covering a larger region.